Consider the second term on the right-hand side of Equation (REF ): the principal minors of the matrix are positive, so the matrix is positive definite if and only if the determinant is positive. This is Sylvester's characterisation of positive definite matrices (see Theorem 7.2.5, [1]}), so Lemma REF  applies. We obtain the inequality
\( \det (G) \le (m-b) \det (G_{n-1}) + b ( m- b )^{n-1},\) 
